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In the long-standing quest for the fundamental building blocks of nature, the so-called Standard Model of particle physics, energy frontier colliders have played a central role in the forefront research for matter and interactions. For future high-energy particle colliders to explore physics beyond the Standard Model, a proton-proton circular collider at energy of 100 TeV in a 100 km circumference or electron-positron linear collider with energy of the order of 1 TeV in a 30 km length is being considered around the world, exploiting the conventional technologies such as superconducting magnets or RF systems . In contrast to proton colliders that create clouds of debris, electron-positron colliders enable cleaner and more precision experiments of fundamental particle collisions. Nowadays, a diversity of electron-positron linear colliders is proposed as a potential application of advanced accelerator concepts , such as two beam accelerators, dielectric wakefield accelerators, beam-driven plasma wakefield accelerators, and laser-driven plasma wakefield accelerators , promising with much higher accelerating gradients than that of a conventional RF accelerator.
Laser-plasma accelerators (LPAs) [4, 5] can support a wide range of potential applications requiring high-energy and high-quality electron-positron beams. In particular, field gradients, energy conversion efficiency, and repetition rates are essential factors for practical applications such as compact free electron lasers [6, 7] and high-energy frontier colliders [8, 9]. Although LPAs provide enormous accelerating gradients, as high as 100 GV/m at the plasma density of 1018 cm−3, dephasing of relativistic electrons with respect to a correct acceleration phase of the plasma wakefield with the phase velocity that is smaller than the speed of light in vacuum, and energy depletion of the laser pulse limit the electron energy gain in a single stage. A straightforward solution to overcome the dephasing and pump depletion effects is to build a multistage accelerator comprising consecutive LPA stages  such that a final energy gain reaches the requirement of the beam energy without loss of the beam charge and qualities through a coupling segment where a fresh laser pulse is fed to continuously accelerate the particle beam from the previous stage. The propagation of laser pulses in plasmas is described by refractive guiding, in which the refractive index can be modified from the linear free space value mainly by relativistic self-focusing, ponderomotive channeling, and a preformed plasma channel . The self-guided LPA [11, 12, 13, 14] relies only on intrinsic effects of relativistic laser-plasma interactions such as relativistic self-focusing and ponderomotive channeling. On the other hand, the channel-guided LPA exploits a plasma waveguide with a preformed density channel [15, 16, 17] or a gas-filled capillary waveguide made of metallic or dielectric materials . The plasma waveguide is likely to propagate a single-mode laser pulse through a radially parabolic distribution of the refractive index and generates plasma waves inside the density channel, the properties of which are largely affected by a plasma density profile and laser power . In contrast with plasma waveguides, the capillary waveguide can guide the laser due to Fresnel reflection on the inner capillary wall, and plasma waves are generated in an initially homogeneous plasma, relying on neither laser power nor plasma density. The presence of the modal structure imposed by the boundary conditions at the capillary wall affects the propagation of a laser pulse through the capillary and thus the excitation of plasma waves inside the capillary. This characteristic allows us to control acceleration of electrons through the modal structure of the propagation of the laser pulse as long as the laser intensity on the capillary wall is kept below the material breakdown [20, 21].
In this paper, we present a novel scheme of a gas-filled capillary accelerator driven by a laser pulse formed from two-mode mixing of the capillary eigenmodes, so-called electromagnetic hybrid modes . Two coupled eigenmodes with a close longitudinal wave number can generate beating wakefields in the capillary. When the beating period is equal to the dephasing distance, the electrons experience the rectified accelerating field; thereby their energy gain can increase over many accelerating phases exceeding the linear dephasing limit and reach the saturation due to the energy depletion of a drive laser pulse in the single-stage LPA. For efficient acceleration of the electron-positron beam up to an extremely high energy such as TeV energies, the multistage accelerator comprising a series of plasma-filled capillary waveguides is a sound approach, in which the particle beam is injected into the initial stage at the right phase of the wakefield from the external injector and accelerated cumulatively in the consecutive accelerating phase of successive stages. For applications of extreme high-energy particle beams to TeV center-of-mass (CM) energy electron-positron linear colliders, minimizing the transverse normalized emittance of the beam particles is of essential importance to meet the requirement of the luminosity of the order of 1034 cm−2 s−1 at 1 TeV CM energy for the particle physics experiments . The numerical model on the bunched beam dynamics in laser wakefields, based on the exact solution of single particle betatron motion taking into account the radiation reaction and multiple Coulomb scattering, reveals that the transverse normalized emittance and beam radius can be consecutively reduced during continuous acceleration in the presence of optimally phased recurrence of longitudinal and transverse wakefields . The final properties of the particle beams reached to the objective energy meet the requirements of the luminosity without resort to an additional focusing system.
The remaining part of this paper is organized as follows. In Section 2, the complete description on the longitudinal and transverse laser wakefields generated by two electromagnetic hybrid modes with moderate intensities coupled to a gas-filled capillary waveguide is provided. In Section 3, the particle beam dynamics on energy gain, beam loading, and betatron motion in a single stage of the two-mode mixing LPA is investigated, taking into account radiation reaction and multiple Coulomb scattering with plasma ions. In Section 4, a multistage coupling with a variable curvature plasma channel is presented. For the multistage comprising two-mode mixing LPAs, the results of numerical studies on the transverse beam dynamics of a particle bunch are shown. Analytical consideration on the evolution of the normalized emittance of the particle beam in the presence of radiation reaction and the multiple Coulomb scattering is given. In Section 5, the performance of a 1-TeV CM energy electron-positron collider comprising the multistage two-mode mixing LPAs is discussed on the luminosity and beam-beam interaction. In Section 6, we conclude our investigation on the proposed laser-plasma linear collider with a summary.
For a large-scale accelerator complex such as the energy frontier particle beam colliders, it is axiomatically useful in assembling a long-range multistage structure for the use of long-term experimental operation at a high-precision and high-repetition rate that each electromagnetic waveguide consists of a simple monolithic structure, as referred to the design of the future electron-positron linear colliders based on radio-frequency technologies . Despite the long-standing research on plasma waveguides comprising density channels generated in plasmas with laser-induced hydrodynamic expansion [23, 24] and pulsed discharges of an ablative capillary [25, 26] or a gas-filled capillary [27, 28], a length of such a plasma channel has been limited to about 10 cm. The pulsed discharge capillaries relying on collisional plasma processes have some difficulties in plasma densities less than 1017 cm−3 and the temporal and spatial stabilities of the density channel properties for the operation at a high repletion rate such as 10 kHz [5, 29]. In contrast to pulsed discharge plasma waveguides, metallic or dielectric capillary waveguides filled with gas [18, 30] will be revisited for a large-scale laser-plasma accelerator operated at a practically higher-repetition rate than 10 kHz, because of the passive optical guiding of laser pulses, the propagating electromagnetic fields of which are simply determined the boundary conditions on a static solid wall of the waveguide unless the laser intensity is high enough to cause the material breakdown on a capillary wall [20, 21]. Furthermore, the modal nature of electromagnetic fields arising from the boundary conditions on a solid wall allows us to conceive a novel scheme that can overcome a drawback of LPAs, referred to as dephasing of accelerated electron beams from a correct acceleration phase in laser wakefields.
Considering the electromagnetic hybrid modes EH1n  to which the most efficient coupling of a linearly polarized laser pulse in vacuum occurs, the normalized vector potential for the eigenmode of the
where is the amplitude of the normalized vector potential defined as for the EH1n mode with the vector potential , the electron charge , electron mass
where is the laser wavenumber with the laser wavelength and is the relative dielectric constant. In the quasi-linear wakefield regime , the ponderomotive force exerted on plasma electrons by two coupled capillary laser fields can be written by , where is defined by averaging the nonlinear force over the laser period , i.e., assuming that in the propagation distance , where is the mode mixing length over which two hybrid modes EH1n and EH1m overlap to cause the beatings of the normalized vector potential, e.g., for the EH11 - EH12 mode mixing of a laser pulse with and in a capillary tube with
The electrostatic potential defined by is obtained from Eq. (5).
where is the plasma frequency. The solution of Eq. (4) is
where is the plasma wavenumber in the capillary, , the mode beating wavenumber and
with the real () and imaginary () part of the error function . For and , the longitudinal electric field generated by the laser pulse can be obtained from as
The transverse focusing force is obtained from as
where is the Bessel function of the first order.
The proposed scheme restricts the laser intensity such that the plasma response is within the quasi-linear regime, i.e., , for two reasons. The one is avoidance of the nonlinear plasma response such as in the bubble regime, where symmetric wakefields for the electron and positron beams cannot be obtained for the application to electron-positron colliders [8, 9] and the degradation of the beam quality due to the self-injection of dark currents from the background plasma electrons. The other is an inherent demand that the laser intensity guided in a capillary tube should be lower enough than the threshold of material damage on the capillary wall .
The coupling efficiency defined by an input laser energy with a spot radius and amplitude coupled to the E1n mode in the capillary with the radius , i.e., is calculated for a linearly polarized Airy beam,
and for a Gaussian beam,
where is the first root of the equation of , as shown in Figure 1a and b, respectively, as a function of . In Eq. (5), the beating term can be maximized by setting at which has the maximum value and the minimum fraction of higher-order modes. As shown in Figure 1, the Airy beam generates the maximum EH11-EH12 mode mixing with and a fraction of higher-order modes with ∼0.5% at , where the coupling efficiencies are , , , , and . The Gaussian beam can generate the EH11-EH12 mode mixing with and a fraction of higher-order modes with ∼5.1% at , where the coupling efficiencies are in the order of , , , , and .
The radial intensity profiles for the EH11, EH12 monomode and EH11-EH12 mixing mode for the Airy beam case are illustrated in Figure 1c–e, respectively. As shown in Figure 1e, a high-intensity region of the mixing mode is confined within a half radius of the capillary, compared to the monomode intensity profiles, which have a widespread robe toward the capillary wall. A centrally concentrated intensity profile of the mixing mode considerably decreases the energy flux traversing on the capillary wall. The normalized flux for EH1n mode at the capillary wall depends on the azimuthal angle as , defined by the ratio of the radial component of the Poynting vector at to the longitudinal component of the on-axis Poynting vector . For the Airy beam with coupled to the capillary with and , the maximum normalized fluxes for the EH11, EH12 mono- and EH11-EH12 mixing modes at or are , , and , respectively. The energy fluence traversing the capillary wall can be estimated by for the peak intensity () and the pulse duration , providing the maximum fluences 19, 66, and 19 mJ/cm2 for the corresponding modes, as shown in Figure 1f. The experimental study of laser-induced breakdown in fused silica (SiO2)  suggests that the fluence breakdown threshold is scaled to be for . According to a more detailed study of laser propagation in dielectric capillaries under non-ideal coupling conditions , the threshold intensity for wall ionization is obtained as () at the wavelength for the capillary radius .
The coupling efficiency of an incident laser pulse to a capillary tube filled with plasma can be improved by the use of a cone-shape entrance of the capillary , suppressing self-focusing effects and increasing the accelerating wakefield excited in the capillary. For the propagation of a laser beam with an approximately Gaussian intensity profile , the evolution of a normalized spot radius can be obtained from the equation , where is the vacuum Rayleigh length, the laser power, and the critical power for relativistic self-focusing with . For the coupling of an Airy beam (or a Gaussian beam) with the radius () to the capillary tube filled with plasma at the electron density of , the cone with the opening radius of () and length () can effectively guide and collect the incident laser energy. The effect of the relativistic self-focusing is estimated by considering the modulation of the refractive index for the EH1n mode, i.e., , where  and . The maximum modulation due to the relativistic self-focusing effect is at most 0.5% for the propagation of the EH11-EH12 mixing modes in a capillary.
In the linear wakefields excited by two coupled modes EH11 and EH12 in the capillary waveguide, the longitudinal motion of an electron traveling along the capillary axis at a normalized velocity is described as .
where is the electron energy, the accelerating field at , the nonrelativistic wave-breaking field, the particle phase with respect to the plasma wave, and . Here, the phase-matching condition is determined such that the beating wavelength is equal to the dephasing length, i.e.,
Taking into account and for and setting the pulse duration of a drive laser pulse with a Gaussian temporal profile to be the optimum length , the on-axis accelerating field near the matching condition is given by
where is a phase mismatching.
While propagating through plasma and generating wakefields, the laser pulse loses its energy as  where is the characteristic scale length of laser energy deposition into plasma wave excitation, referred to as the pump depletion length. In the linear wakefield regime where a laser pulse duration is assumed to be fixed, the laser energy evolution in the capillary can be written as , taking into account the energy attenuation of two coupled hybrid modes. In the quasi-linear wakefield regime, i.e., , the scaled pump depletion length is given by with for a Gaussian laser pulse [9, 37], while the scaled coupled mode attenuation length yields with the matching condition given by Eq. (12), i.e., for and the glass with the relative dielectric constant . Hence, the damping of wakefields during the laser pulse propagation is dominated by the energy depletion of the laser pulse as given in Eq. (13). Thus, integrating the equations of motion in Eq. (11) over , the energy and phase of the electron can be obtained as
where is the initial electron energy, the initial electron phase with respect to the wakefield, the maximum electron phase in the wakefield for the matching condition in Eq. (12) and the laser pulse length , and
The maximum energy gain to be attainable at is obtained as
Considering the mixing of two lowest order hybrid modes EH11 and EH12 with the coupling efficiencies and , the evolution of the energy gain with respect to is shown in Figure 2a for various detuning phases in comparison with that of the EH11 monomode with and . The effect of phase mismatching on the maximum attainable energy gain is shown in Figure 2b for various normalized laser intensities in the quasi-linear regime. One can see that the growth of energy gain occurs in the relatively wide range of the phase mismatching over and that the maximum attainable energy gain does not strongly depend on the normalized vector potential in the quasi-linear regime. While the single-mode LPA driven by the normalized intensity reaches the maximum energy gain over the accelerating phase , the two-mode mixing LPA with the phase matching, i.e., , is attainable to the maximum energy gain over the accelerating phase region , as shown in Figure 2. It is noted that significant enhancement of the energy gain is attributed to a large energy transfer efficiency from the laser pulse to the wakefield, i.e., over the accelerating phase region , while the energy transfer efficiency for the single-mode LPA is over the accelerating phase region .
The average energy gain of electrons contained in the bunch with the root-mean-square (rms) bunch length and longitudinal Gaussian density distribution can be estimated as , where
Figure 3 shows the evolution of the energy gain and the maximum attainable energy gain averaged over electrons in a Gaussian bunch with various rms lengths. It is noted that the maximum attainable energy gain at exhibits weak dependence on the initial bunch phase for a long bunch and that the minimum energy spread occurs at for different bunch lengths.
In the linear regime, a solution of the Green’s function for the beam-driven wakefield excited by a charge bunch with bi-Gaussian density distribution , i.e., and for the rms bunch length , rms bunch radius , and particle charge ( for a positron beam and for an electron beam), is written as , where is the coordinate in the co-moving frame of a relativistic electron beam with and is the radial, transverse coordinate of an electron beam having a cylindrical symmetry . Here, the longitudinal and transverse plasma responses are obtained as
and inside the bunch ()
where is the modified Bessel function of the second kind and is the incomplete Gamma function of the second kind. Combining the longitudinal and transverse solutions, the wakefield excited by a bi-Gaussian-shaped bunch is obtained as
where is the electron classical radius, the number of particles in a bunch, and . If we consider a laser-driven wakefield excited by two mixing hybrid modes accelerating the electron beam in a gas-filled capillary, the net longitudinal electric field, i.e., the beam loading field, experienced by the electron beam is given by . From Eqs. (13) and (20), the beam loading field at consisting of the laser- and beam-driven wake, where the electron bunch is located at in the laser co-moving frame, i.e., , yields
where . A loss of the energy gain due to the beam wakefield at the bunch center is
and the rms energy spread due to the beam loading is estimated as